The purpose of the proposed work is to obtain new empirical information and theoretical analyses about thinking and problem solving. The idea to be used is that reasoning involves a process of carrying out a series of transformations on cognitive structures. For example, in solving a problem, a person begins with some cognitive representation of the problem, then transforms that cognitive structure in various ways until the solution has been found. The main theoretical proposal is that we can represent a significant variety of these cognitive transformations as geometric and topological operations -- that is, we can treat geometry and topology as analogues of "cognitive space" and analyze cognitive transformations in relation to geometric and topological transformations. Experiments in which this idea will be worked out will include studies of spatial analogies, serial pattern learning and serial completion tasks based on spatial transformations, and studies of problems solving in geometry including proof of theorems.